Let k≥ 3 and n≥ 3 be odd integers, and let m≥ 0 be any integer. For a prime number ℓ, we prove that the class number of the imaginary quadratic field Q(ℓ2m-2kn) is either divisible by n or by a specific divisor of n. Applying this result, we construct an infinite family of certain tuples of imaginary quadratic fields of the form: (Q(d),Q(d+1),Q(4d+1),Q(2d+4),Q(2d+16),⋯,Q(2d+4t))with d∈ Z and 1 ≤ 4 t≤ 2 | d| whose class numbers are all divisible by n. Our proofs use some deep results about primitive divisors of Lehmer sequences.